Most mesh-based tracking/modeling techniques face a point at which they need to approximate the underlying image function with a smoother, more regular function. This often occurs when gradients need to be computed. The typical approach is to convolve the image with a fixed kernel. This has a number of disadvantages. First, a fixed kernel does not account for the different levels of detail in different parts of the mesh – small facets can be oversmoothed while large facets are undersmoothed. Also, the size of the tracked image may be changing dramatically in scale (as in head tracking), and thus a fixed-size kernel again presents problems. We present a method to approximate a function with a set of basis functions that are consistent with the mesh (planar triangular patches above each facet). The resulting approximation is piecewise analytic (with C0 continuity) and inherently at the same level of detail as the mesh. We show how this approximation can be very efficiently computed, touching each pixel only once, and how multiscale representations can be computed with minimal cost. We describe techniques and applications for image coding, gradient computation, tracking, and adaptive remeshing. Examples of image functions, object meshes, and the resulting approximations are shown.